Place Chicken Stock, Chicken, Onion, Carrots and Garlic in a large pot and bring to a boil. Once boiling allow to simmer on low for 45 minutes.
Remove Chicken from soup pot and set aside. Remove soup from burner and using an immersion hand blender or blender, blend the chicken stock, carrots, onion and garlic until smooth.
If using a blender, be sure to be very careful to avoid hot soup going everywhere. When blending, the soup will expand, therefore you will want to remove the center cap from the lid, only fill the blender half way, and hold a folded hand towel over the opening to catch any flying soup. You may need to work in batches. Once all soup is blended return the soup back to the pot. I love my immersion hand blender and use it all the time. If you don't have one, I highly recommend them!
Once the soup is blended turn the stove back on to medium-low heat and add the Cream of Chicken and Cheddar Cheese soup to the stock mixture.
Cut the Velveeta into dice sized chunks. Carefully drop the chunks into the soup one by one. You don't want to dump them in all at once because they will sink to the bottom and easily burn. Keep stirring until the cheese has fully melted and there are no chunks left.
Shredded the chicken breasts using two forks, then add back to the soup.
Add the broccoli and cook until broccoli is cooked through.
Notes
I love to serve this soup with warm bread or rolls.